Due to the additional need of very short bunches for the FEL operation with the TESLA-machine strong wakefield effects are expected. One third of the total wakefield energy per bunch is radiated into the frequency region above the energy gap of Cooper pairs in superconducting niobium. The energy of the cooper pairs in superconducting niobium at 2 K corresponds to a frequency of 700 GHz. An analytical and experimental estimation for the overall energy loss of the FEL bunch above energy gap is presented. The analytical method is based on a study from R. B. Palmer [1]. The results of the wakefield estimations are used to calculate possible quality factor reduction of the TESLA cavities during FEL operation. Results are presented

Hülsmann, Peter

Klein, Horst

Müller, Wolfgang F. O.

Peschke, Claudius

English

Hochschulschriftenserver - Universität Frankfurt am Main

T T T TH H H HE E E E       III IN N N NF F F FL L L LU U U UE E E EN N N NC C C CE E E E       O O O OF F F F       W W W WA A A AK K K KE E E EF F F FIII IE E E EL L L LD D D DS S S S       O O O ON N N NS S S SU U U UP P P PE E E ER R R RC C C CO O O ON N N ND D D DU U U UC C C CT T T TIII IN N N NG G G G        T T T TE E E ES S S SL L L LA A A A        C C C CA A A AV V V VIII IT T T TIII IE E E ES S S S        III IN N N NF F F FE E E EL L L L- - - -O O O OP P P PE E E ER R R RA A A AT T T TIII IO O O ON N N NP. H￿LSMANN, H. KLEIN, C. PESCHKE, W. F. O. M￿LLERInstitut f￿r Angewandte Physik derJohann Wolfgang Goethe-Universit￿tRobert-Mayer-Str. 2-4, D-60054 Frankfurt am Main, FRGA A A Ab b b bs s s st t t tr r r ra a a ac c c ct t t tDue to the additional need of very short bunches for the FEL operation withthe TESLA-machine strong wakefield effects are expected. One third of thetotal wakefield energy per bunch is radiated into the frequency region abovethe energy gap of Cooper pairs in superconducting niobium. The energy of thecooper pairs in superconducting niobium at 2 K corresponds to a frequency of700 GHz. An analytical and experimental estimation for the overall energyloss of the FEL bunch above energy gap is presented. The analytical methodis based on a study from R. B. Palmer [1]. The results of the wakefieldestimations are used to calculate possible quality factor reduction of theTESLA cavities during FEL operation. Results are presented.1 1 1 1. . . .        III In n n nt t t tr r r ro o o od d d du u u uc c c ct t t tiii io o o on n n nTo drive coherent X-ray sources with TESLA, the acceleration of long trainsof very short and intense electron bunches up to 50 GeV is necessary. Boththe longitudinal and transversal emittance growth in the linac is required tobe kept small. Furthermore the heating of the cavity walls due to wakefieldsin the high frequency regime has to be limited. Long range wakefields drivenby HOMs (Higher Order Modes) are one of the primary sources of multi-bunchemittance growth. But those HOMs can be suppressed using HOM-couplers.Short range wakefields cannot be avoided. They still increase the singlebunch energy spread and the emittance of the beam. A crucial point in theTESLA-FEL study is the requirement of extremely short bunches. Those short 1bunches generate wakefields far into the THz regime which may have thecapability to destroy Cooper pairs.The TESLA-FEL study [2] foresees the acceleration of 11315 bunches withina time of 800 ms with a spacing of 70 ns from bunch to bunch. It considers abunch length of 0.083 ps and a peak current of 13.4 kA. From the short bunchlength arise severe problems for computer simulations of wakefields.Therefore we have applied the analytical method of R. B. Palmer [1] tocalculate the overall energy losses of a very short bunch passing the TESLA-channel. This analytical method is of course only an estimation but it isadvantageous with regard to the physical insight into the complicatewakefield behaviour. It delivers the single cell solution as well as thesolution for a periodic chain of cells. For short bunches the wakefieldsconverge after a large number of cavities to a periodic solution which leadsto energy losses being much smaller than in the single cell case.The frequency which corresponds to the energy gap of Cooper pairs inniobium at 2 K is 700 GHz. The amount of energy above the energy gapradiated into the shadow region of the cavity was calculated with themethods of optical diffraction theory.The heating of the cavity surface was calculated by solving the timedependent heat conduction equation under consideration of the temperaturedependence of the heat conductivity.2 2 2 2. . . .        E E E En n n ne e e er r r rg g g gy y y y        L L L Lo o o os s s ss s s s        iii in n n n        t t t th h h he e e e        T T T TE E E ES S S SL L L LA A A A        C C C Ch h h ha a a an n n nn n n ne e e elll lFor very short bunches the behaviour of the impedance and loss factor isdominated by the fields diffracted at the cavity edges. Most of the energy isradiated in the frequency region far above the cut off frequency of the beampipe.Consider a bunch of charge Q and rms size s passing a pill box cavity withbeam pipes. The field excited by the head of the bunch and diffracted by thefirst edge, touches the bunch tail if (see Fig.1). (1)During the bunch passage the self field has expanded into the shadow regionof the cavity. For z=g the radius of the self field is larger approximately by 2the amount. (2)The expansion of the self field is shown in Fig. 1. The field is modifiedwithin the region a-d < r < a+d.zÖz2+ 2gaFig. 1: Expansion of the self field of an electron bunch passing a single pill boxNow we assume that the decrease of the energy density   dU / dris approximately linear with the radius r and has a negligible value by a+d(see Fig. 2). Thus the energy within the bunch length in the field outsideradius a will be given by (for d << a). (3) 3If the cavity ends after the distance g, all the energy outside the beam piperadius a will be reflected at the wall and will be lost to the bunch.Consequently equation (3) gives the energy lost by the beam, except for afactor of two. According to the standard diffraction theory there is also aloss into diffractive fields that propagate down the beam pipe in addition tothe lossra1Fig. 2: Energy loss in one pill box cavityinto the cavity shadow region. These two parts of the diffracted fields carryequal energy [6]. If we replace d by expression (2), the total energy loss willbe. (4)For a pill box with the fundamental parameters of the TESLA-geometry(g=115 mm, a=35mm) and a bunch length of 25 mm equation (4) gives. (5)To calculate the energy loss in multiple cavities we consider a sequence ofn identical cavities each separated by vertical thin irises at a spacing of g.The expansion of the self field into the shadow region of the nth cell is thesame we calculated in equation (2). But inside of the iris radius the extent 4of the disturbance increases with the number of passed cells and thus oneobtains. (6)ra1i iFig. 3: Energy loss in multiple pill box cavities.The total distance over which the  energy density within the bunch lengthmust fall is. (7)Again we assume the field energy to decrease linearly between a-di and a+d,then the energy loss of the bunch within the nth cell will be. (8)We see that the loss per cavity decreases with increasing number ofcavities. To calculate the overall energy loss within the TESLA-9-cellstructure we have to sum up the contributions from cell 1 to cell 9. We thenhave   DUTESLA –9–cell = DUPill Box2(1 + n) Sn =19» 6 DUPill Box » 78 mJ. (10) 5The energy loss is commonly expressed by a loss factor which is defined by    k = DUQ2.Here one finds. (11)According to equation (9) the energy loss goes asymptotically against zeroif the number of passed cavities goes against infinity. But that is notpossible. At a certain number of n the energy loss will settle to a quasiequilibrium. In that case the modification of the field pattern has reachedthe bunch (see Fig. 4). The approximate number of cavities ncrit before thequasi equilibrium state is reached can be calculated by equation (6) if wereplace di by the iris radius. Then we get   nequilibrium= a22 g s » 213. (12)renergy density atequilibrum state(after cell #213)a1Fig. 4: Energy loss after cell number 213 (quasi equilibrium state).The magnitude of the energy loss per cell in the equilibrum case will beobtained by substituting a for di in equation (8). From cell number 213 on on 6one has to calculate the energy loss by   DUTESLAequilibrium=Q22 pe0 a2 L » 19,1 mJ (13)per 9-cell-structure. This leads to the corresponding loss factor per meter which is then given by. (14)The theory described above may be compared with a numerical simulation ofthe loss factor for a TESLA 9 cell structure and a bunch length of 1 mm.Reference [5]  gives a value for the logitudinal loss factor of.From equation (10) the loss factor is,which is in reasonable agreement with the numerical calculation.3 3 3 3. . . .        T T T Th h h he e e e        E E E En n n ne e e er r r rg g g gy y y y        L L L Lo o o os s s ss s s s        a a a ab b b bo o o ov v v ve e e e        7 7 7 70 0 0 00 0 0 0        G G G GH H H Hz z z zWith the methods of optical diffraction theory one can calculate the energyloss of a point like bunch passing through a single cavity with beam tubes[3]. This method can be used to obtain a reasonable estimate of the highfrequency part of the energy loss in a cavity. If, however, we only know thehigh frequency part of the energy loss we still can calculate the fraction ofenergy which is radiated into the high frequency region beyond 700 GHz.According to the diffraction theory the energy loss of a cavity at highfrequencies is, (15) 7where Z0=377 W, the impedance of free space. The subscript g denotes theenergy loss beyond the threshold energy for braking Cooper pairs. With (16)Fig. 5: The diffraction geometry [3] for one wavelength. J and J0 being thetime-averaged Poynting vector for the perturbed and unperturbedwave respectively.we find for the bunch length of 25 mm and the threshold frequency fg=700GHz, that 33% of the overall energy loss is radiated into the frequency 8region beyond the threshold energy. Thus we have for a TESLA-9-cell-structure, (17)and in the equilibrium case beyond cell number 213(18)energy loss into the shadow region of the cavity beyond the thresholdenergy. Please note, that the energy deposited in the equilibrium case issmaller by a factor of 4 than in the nonequilibrium case. At the beginning apower of(19)is radiated into the shadow region of a TESLA 9 cell structure beyond thethreshold energy during one macropulse of 800 ms length with 11315bunches spaced by 70 ns. In the equilibrum case we only get. (20)4 4 4 4. . . .        T T T Th h h he e e e        P P P Po o o ow w w we e e er r r r        D D D Diii is s s ss s s siii ip p p pa a a at t t te e e ed d d d        iii in n n n        t t t th h h he e e e        C C C Ca a a av v v viii it t t ty y y y        W W W Wa a a alll llll lFor the discussion below let us consider the wakefields as photons, whichseems reasonable because of the high frequencies of the wakefields. Asecond argument for doing so is the loss of correlation after somereflections due to the roughness of the niobium surface.Now we have to answer the question:How much power being radiated into the shadow region of the cavity canescape through the beam pipes ?On the first view the question seems simple but it is not. For example onecan try to calculate numerically the probability, that a photon which wasborn in a certain cell will leave the cavity, using something like a particletracking code. This must fail because of the elliptical shape of the TESLA-cavity surface. We expect a chaotic scattering of photons. Consequently we 9tried to solve the problem approximately by a simple experiment. A lightbulb was inserted in the shadow region of one cell within the TESLA-9-cell-structure. During one measurement the position of the light bulb remainsfixed. For the next measurement the bulb was moved to the next cell and soon. The power of light which enters the beam pipes to the left and the rightof the cavity was detected by a calibrated solar cell. During onemeasurement the solar cell was mounted to one beam pipe whereas the otherbeam pipe was closed by a black box to avoid reflections.Fig. 6: The light bulb experimentOne can see by the dotted curves in Fig. 7 that only the first two and the lasttwo cells contribute significantly to the power loss through the beam pipes.The contribution of the inner cells are very small and can be neglected.Therefore the power loss caused by photons entering the beam pipes is inthe order of 5 %.Of course the reflection factors of superconducting niobium and normalconducting copper are different and thus this measurement is only anapproximation. For the first 9-cell-cavity we have. (21)In the equilibrium case beyond cell number 213 we find. (22) 1010-50,00010,0010,010,110123456789 1 0probabilitycell numberFig. 7:  Probability that a photon enters the beam pipe in logarithmic scaleversus cell number. During one measurement the position of the solarcell and the black box where exchanged from the left beam pipe tothe right beam pipe. Thus we measured two curves.This is additional power which has to be dissipated in one 9-cell-cavity. Onemodule is made of 8 cavities and the duty cycle of TESLA-FEL is 0.526%. Theadditional dissipated power for one module in the equilibrium state is:    DP =4 4W × 8 × 0.00526 = 1.852 WThe power dissipated at 2 K from all other sources is 10.81 W per moduleand the high frequency wakes above 700 GHz cause an additional term (inworst case) of17.1 % 115 5 5 5. . . .        H H H He e e ea a a at t t tiii in n n ng g g g        o o o of f f f        t t t th h h he e e e        T T T TE E E ES S S SL L L LA A A A- - - -C C C Ca a a av v v viii it t t ty y y y        d d d du u u ur r r riii in n n ng g g g        O O O On n n ne e e e        M M M Ma a a ac c c cr r r ro o o op p p pu u u ulll ls s s se e e eIn order to calculate the increase of temperature of the inner cavitysurface, the stationary heat conduction equation was numerically solvedunder consideration of the temperature dependence of the heat conductivity(see Fig. 6). Therefore the niobium layer was divided into thin sheets withconstant heat conductivity.The calculations where performed with the following boundary conditions:1) The temperature of the outer cavity surface was kept constant and equalto 2 K.2) The acceleration gradient of 25 MV/m was kept constant. Thus from thedecreasing quality factor follows an increasing power loss to maintainthe acceleration gradient. The unloaded quality factor was assumed to be5*109.In a first step we have calculated the time constant of the heating, in orderto show, that the stationary heat conduction equation is applicable.Therefore we have solved the time dependent heat conduction equation    ¶DT(x,t)¶t=l wr c¶DT(x,t)¶x, (23)approximately, where lw, r and c are the heat conductivity, the density andthe specific heat of the niobium. The meaning of DT is    DT(x,t)=T(x,t)–THe(x,t) . (24)At x=0 we have the HF-surface and at x=d the outer surface is in contactwith the fluid helium at 2 K. For the calculation of the time constantequation (23) was solved by the well known ansatz of a Fourier series withthe assumption, that lw, r and c are constants. We found   T(x,t)= THe + DT 4p2 {1–(–1)n}cos n p2 dx e–a n t Sn =1¥, (25) 12with    a n =l wr c (n p2 d)2 . (26)Due to the fact, that only the first term is dominating, we have    t =r cl w(2 dp )2. (27)With the assumptions   c » 7,01 × 10–3 Jkg K, r»8580kgm3, l w » 4,25jmsK , (28)the time constant of the heating is approximately    t»40 ms . (29)This is relatively small compared to 800 ms pulse duration time. Thereforeone can calculate the increase of temperature by  the methods of stationaryheat conduction equation. According to Fig. 8 the temperature dependence ofthe heat conductivity was assumed to be    l w » 4,25 × ( TTHe)3,9. (30)The temperature dependence of the quality factor was considered by theformula   Q(T)=Q(THe) TTHee– Dk( 1THe– 1T), (31)which can be calculated in the frame of BCS-theory. 136 6 6 6. . . .        R R R Re e e es s s su u u ulll lt t t ts s s sThe final temperature of the inner cavity surface is reached withinapproximately 80 ms.For the increase of temperature we found the following results (see Fig. 9and Fig.10):2.4 K for the first cavity. The corresponding quality factor is 1.47*109(compared to Q0 = 5*109) and the additional needed power for theacceleration mode is 358 W.2.26 K for the cavities beyond cavity number 213. The corresponding qualityfactor is 2.2*109 and the additional needed power is 169 W.These results we obtained with the assumption, that the increase oftemperature is caused only by the wake power beyond 700 GHz .If we further take into an account the whole wake power, which is of coursethe worst case (most of the wake power being removed by the HOM-couplers), we found:2.7 K for the first cavity. The corresponding quality factor is 0.77*109 andthe additional needed power is 1.21 kW.2.36 K for the cavities beyond cavity number 213. The corresponding qualityfactor is 1.64*109 and the additional needed power is 442 W.Fig. 11 shows the temperature profile inside the niobium layer at differentdissipated power levels.7 7 7 7. . . .        C C C Co o o on n n nc c c clll lu u u us s s siii io o o on n n ns s s sWith the assumption, that the increase of temperature is caused by the wakepower beyond 700 GHz only, the FEL-operation seems to be possible. Thequality factor of the first cavities is reduced by a factor of 3.4 and beyondcavity number 213 by a factor of 2.3 .If the whole high frequency wake power contributes to the increase oftemperature, the FEL-operation turned out to be critical.For all calculations Palmers theory on wakefields of very short bunches wasadopted. 14110100100024 6 8 1 0 3 0l[ W / m · K ]T [K]RRR = 380RRR = 1070Fig. 8: Heat conductivity versus temeperature for niobium. 152K2,1K2,2K2,3K2,4K2,5K2,6K2,7K2,8K2,9K0 W 500W 1000W 1500W 2000WPHF+PWakeTFig. 9: Temperature at the inner surface versus the power dissipated in a nine cell cavity (RRR=300).10810910101011101210131K 2K 3K 4K 5K 6KTQ2.26K 2.70K5·1091.64·1097.74·1081.47·1092.2·1092.36K 2.40KFig. 10: Quality factor versus temeperature. The unloaded Q is assumed to be 5*109 (RRR=300). 162K2,1K2,2K2,3K2,4K2,5K2,6K2,7K0mm 0,5mm 1mm 1,5mm 2mm 2,5mmToutersurfaceinnersurfaceP=1388WP=532WP=343WP=616WFig. 11: Temperature profile inside the niobium layer for a 9 cell TESLA      cavity at different dissipated power levels.R R R Re e e ef f f fe e e er r r re e e en n n nc c c ce e e es s s s1. R. B. Palmer, ￿A Qualitative Study Of Wake Fields For Very ShortBunches,￿ Particle Accelerators, 1990, Vol. 25, pp. 97-1062. R. Brinkmann, ￿On The Free Electron Laser Mode Operation InTESLA,￿TESLA Report, Nr. 96-01, January 19963. K. Bane and M. Sands, ￿Wakefields Of Very Short Bunches In An Accelerating Cavity,￿ Particle Accelerators, 1990, Vol. 25, pp. 73-954. R. Brinkmann et al., Conceptual Design of a 500 GeV e+e- Linear Collider with Integrated X-Ray Laser Facility5. A. Mosnier, ￿Longitudinal and Transversal Wakes for the TESLA Cavity,￿TESLA Report, Nr. 93-11, May 1993 17

The influence of wakefields on superconducting TESLA-cavities in FEL-operation

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